Solutions of Linear Equation in Two Variables
A linear equation in two variables is an equation of the form ax + by + c = 0, where a, b, and c are real numbers and both a and b are not zero simultaneously.
Unlike linear equations in one variable (which have a unique solution), a linear equation in two variables has infinitely many solutions. Each solution is an ordered pair (x, y) that satisfies the equation.
In Class 9 Mathematics, students learn to find multiple solutions, express them as ordered pairs, verify whether a given pair is a solution, and represent the solution set graphically as a straight line on the Cartesian plane.
What is Solutions of Linear Equation in Two Variables?
Definition: A solution of a linear equation in two variables ax + by + c = 0 is an ordered pair (x₀, y₀) such that ax₀ + by₀ + c = 0.
ax + by + c = 0 has infinitely many solutions
Where:
- a and b are the coefficients of x and y respectively (not both zero)
- c is the constant term
- Each solution is an ordered pair (x, y)
- The set of all solutions forms a straight line when plotted on the Cartesian plane
Important:
- A linear equation in two variables does NOT have a unique solution — it has infinitely many.
- To find a solution, assign any value to one variable and solve for the other.
- Every point on the graph of the equation is a solution.
- Every solution of the equation corresponds to a point on the graph.
Solutions of Linear Equation in Two Variables Formula
Key Formulas and Methods:
1. Standard Form:
ax + by + c = 0 (a ≠ 0 or b ≠ 0)
2. Finding y in terms of x:
- If b ≠ 0: y = (−ax − c) / b
- Choose any value of x, substitute, and calculate y.
3. Finding x in terms of y:
- If a ≠ 0: x = (−by − c) / a
- Choose any value of y, substitute, and calculate x.
4. Verification of a Solution:
- Substitute the given values of x and y into the equation.
- If LHS = RHS, the pair is a solution.
- If LHS ≠ RHS, the pair is NOT a solution.
5. Number of Solutions:
- One equation in two variables → infinitely many solutions.
- Two independent equations in two variables → exactly one solution (if consistent).
Derivation and Proof
Why a Linear Equation in Two Variables Has Infinitely Many Solutions:
Step 1: Consider the equation
- Take the equation 2x + y = 10.
Step 2: Express y in terms of x
- y = 10 − 2x
Step 3: Substitute different values of x
- x = 0 → y = 10 − 0 = 10 → Solution: (0, 10)
- x = 1 → y = 10 − 2 = 8 → Solution: (1, 8)
- x = 2 → y = 10 − 4 = 6 → Solution: (2, 6)
- x = 3 → y = 10 − 6 = 4 → Solution: (3, 4)
- x = 5 → y = 10 − 10 = 0 → Solution: (5, 0)
- x = −1 → y = 10 + 2 = 12 → Solution: (−1, 12)
- x = 0.5 → y = 10 − 1 = 9 → Solution: (0.5, 9)
Step 4: Conclusion
- Since x can take any real value (there are infinitely many real numbers), there are infinitely many corresponding values of y.
- Each (x, y) pair is a valid solution.
- Geometrically, these infinitely many points form a straight line.
Types and Properties
Types of Solutions and Special Cases:
1. General Solutions
- For ax + by + c = 0 with b ≠ 0: choose any x, compute y = (−ax − c)/b.
- Each choice gives a different solution.
- The solution can involve fractions and negative numbers.
2. Integer Solutions
- Sometimes the problem asks for solutions where both x and y are integers.
- Example: For 3x + 2y = 12, integer solutions include (0, 6), (2, 3), (4, 0).
3. Natural Number Solutions
- Solutions where both x and y are positive integers (natural numbers).
- These are finite in number, even though the total solution set is infinite.
- Example: For x + y = 5, natural number solutions: (1, 4), (2, 3), (3, 2), (4, 1).
4. Solutions on the Axes
- Setting x = 0 gives the y-intercept: (0, −c/b).
- Setting y = 0 gives the x-intercept: (−c/a, 0).
- These two solutions are often the easiest to find.
5. Trivial Form y = mx
- When c = 0, the equation becomes ax + by = 0 or y = (−a/b)x.
- The line passes through the origin (0, 0).
- (0, 0) is always a solution.
Solved Examples
Example 1: Example 1: Find four solutions of 2x + y = 7
Problem: Find four solutions of the equation 2x + y = 7.
Solution:
Express y in terms of x: y = 7 − 2x
- x = 0 → y = 7 − 0 = 7 → (0, 7)
- x = 1 → y = 7 − 2 = 5 → (1, 5)
- x = 2 → y = 7 − 4 = 3 → (2, 3)
- x = 3 → y = 7 − 6 = 1 → (3, 1)
Answer: Four solutions are (0, 7), (1, 5), (2, 3), (3, 1).
Example 2: Example 2: Verify whether (2, 3) is a solution of 3x − 2y = 0
Problem: Check if the point (2, 3) is a solution of 3x − 2y = 0.
Solution:
Substitute x = 2, y = 3 in the equation:
- LHS = 3(2) − 2(3) = 6 − 6 = 0
- RHS = 0
- LHS = RHS
Answer: Yes, (2, 3) is a solution of 3x − 2y = 0.
Example 3: Example 3: Check if (1, 2) is a solution of x + 3y = 8
Problem: Is (1, 2) a solution of x + 3y = 8?
Solution:
Substitute x = 1, y = 2:
- LHS = 1 + 3(2) = 1 + 6 = 7
- RHS = 8
- LHS ≠ RHS
Answer: No, (1, 2) is NOT a solution of x + 3y = 8.
Example 4: Example 4: Express an equation and find solutions
Problem: The cost of a notebook is twice the cost of a pen. If a pen costs ₹x and a notebook costs ₹y, write a linear equation and find three solutions.
Solution:
Forming the equation:
- Cost of notebook = 2 × cost of pen
- y = 2x, or equivalently, 2x − y = 0
Finding solutions:
- x = 5 → y = 10 → (5, 10)
- x = 10 → y = 20 → (10, 20)
- x = 15 → y = 30 → (15, 30)
Answer: Equation: 2x − y = 0. Three solutions: (5, 10), (10, 20), (15, 30).
Example 5: Example 5: Find the value of k if (2, k) is a solution
Problem: Find the value of k if (2, k) is a solution of 3x + 4y = 22.
Solution:
Substitute x = 2, y = k:
- 3(2) + 4k = 22
- 6 + 4k = 22
- 4k = 22 − 6 = 16
- k = 16 / 4 = 4
Verification: 3(2) + 4(4) = 6 + 16 = 22 = RHS ✓
Answer: k = 4
Example 6: Example 6: Find integer solutions of x + y = 5 where x, y > 0
Problem: Find all natural number solutions of x + y = 5.
Solution:
y = 5 − x. Both x and y must be positive integers.
- x = 1 → y = 4 → (1, 4)
- x = 2 → y = 3 → (2, 3)
- x = 3 → y = 2 → (3, 2)
- x = 4 → y = 1 → (4, 1)
For x = 0 or x = 5, one value becomes 0 (not a natural number). For x ≥ 5, y becomes 0 or negative.
Answer: There are exactly 4 natural number solutions: (1, 4), (2, 3), (3, 2), (4, 1).
Example 7: Example 7: Write the equation as ax + by + c = 0 and find intercepts
Problem: Write 5x − 3y = 15 in standard form and find the x-intercept and y-intercept.
Solution:
Standard form: 5x − 3y − 15 = 0 (a = 5, b = −3, c = −15)
X-intercept (set y = 0):
- 5x − 0 = 15 → x = 3
- X-intercept = (3, 0)
Y-intercept (set x = 0):
- 0 − 3y = 15 → y = −5
- Y-intercept = (0, −5)
Answer: Standard form: 5x − 3y − 15 = 0. X-intercept: (3, 0). Y-intercept: (0, −5).
Example 8: Example 8: Two solutions to draw a line
Problem: Find two solutions of 4x + 3y = 24 so that the line can be drawn on the Cartesian plane.
Solution:
Solution 1: Set x = 0:
- 3y = 24 → y = 8 → (0, 8)
Solution 2: Set y = 0:
- 4x = 24 → x = 6 → (6, 0)
Plotting: Mark (0, 8) on the y-axis and (6, 0) on the x-axis. Join them with a straight line. Every point on this line is a solution of 4x + 3y = 24.
Answer: Two solutions are (0, 8) and (6, 0).
Example 9: Example 9: Express a word problem as an equation
Problem: Ravi has some ₹5 coins and ₹10 coins. The total amount is ₹80. Write a linear equation and find three solutions.
Solution:
Let:
- x = number of ₹5 coins
- y = number of ₹10 coins
Equation: 5x + 10y = 80 → x + 2y = 16
Finding solutions (x and y must be non-negative integers):
- y = 0 → x = 16 → (16, 0)
- y = 2 → x = 16 − 4 = 12 → (12, 2)
- y = 4 → x = 16 − 8 = 8 → (8, 4)
Answer: Equation: x + 2y = 16. Three solutions: (16, 0), (12, 2), (8, 4).
Example 10: Example 10: Solutions with fractional values
Problem: Find two solutions of 3x + 5y = 1 involving fractions.
Solution:
Express y in terms of x: y = (1 − 3x) / 5
Solution 1: Let x = 2:
- y = (1 − 6) / 5 = −5/5 = −1
- Solution: (2, −1)
Solution 2: Let x = 1/3:
- y = (1 − 1) / 5 = 0/5 = 0
- Solution: (1/3, 0)
Verification of (2, −1): 3(2) + 5(−1) = 6 − 5 = 1 = RHS ✓
Answer: Two solutions are (2, −1) and (1/3, 0).
Real-World Applications
Applications of Linear Equations in Two Variables:
- Cost and Revenue Problems: If an item costs ₹x and another costs ₹y, a total cost equation like 3x + 5y = 100 models the relationship. Each solution gives a valid combination of quantities.
- Mixture Problems: Mixing solutions of different concentrations uses linear equations. For example, mixing x litres of 10% solution with y litres of 30% solution to get a desired concentration.
- Distance-Speed-Time Problems: If two vehicles travel towards each other, the equation d₁ + d₂ = D (total distance) gives a linear equation in two variables (speeds or times).
- Age Problems: Statements like "Father is 30 years older than son" translate to y − x = 30, a linear equation with infinitely many solutions representing all possible age pairs satisfying the condition.
- Geometry — Perimeter and Area: If the length is x and width is y, then 2x + 2y = P (perimeter) is a linear equation whose solutions give all rectangles with that perimeter.
- Data Analysis: Linear trends in data (e.g., y = 2x + 5) represent relationships where each data point is a solution of the equation.
Key Points to Remember
- A linear equation in two variables has the form ax + by + c = 0 where a and b are not both zero.
- It has infinitely many solutions, each represented as an ordered pair (x, y).
- To find a solution, assign a value to one variable and solve for the other.
- The graph of a linear equation in two variables is always a straight line.
- Every point on the line is a solution; every solution lies on the line.
- The x-intercept is found by setting y = 0; the y-intercept by setting x = 0.
- Two solutions are sufficient to draw the graph (a third can be used for verification).
- If c = 0, the line passes through the origin.
- Natural number solutions are finite even though total solutions are infinite.
- An ordered pair (p, q) is a solution if and only if substituting x = p and y = q makes the equation true.
Practice Problems
- Find four solutions of the equation x + 2y = 6.
- Check whether (−1, 3) is a solution of 2x + 5y = 13.
- Find the value of p if (3, p) is a solution of 4x − y = 7.
- Write 7x − 2y = 14 in the form ax + by + c = 0 and find both intercepts.
- A bus charges ₹15 per adult and ₹10 per child. Total collection is ₹300. Write the equation and find three valid (whole number) solutions.
- How many natural number solutions does 2x + 3y = 18 have? List all of them.
- Find two solutions of 5x − 4y = 3 that involve fractions.
- If 3x + ky = 9 has (1, 2) as a solution, find k.
Frequently Asked Questions
Q1. How many solutions does a linear equation in two variables have?
Infinitely many. Since you can assign any real value to one variable and compute the other, there is no limit to the number of solutions.
Q2. What is a solution of a linear equation in two variables?
A solution is an ordered pair (x, y) that satisfies the equation. When x and y are substituted into the equation, the left-hand side equals the right-hand side.
Q3. How do you verify a solution?
Substitute the given values of x and y into the equation. If LHS = RHS, the pair is a valid solution. If not, it is not a solution.
Q4. What is the graph of a linear equation in two variables?
The graph is a straight line. Every point on this line represents a solution of the equation. Conversely, every solution corresponds to a point on the line.
Q5. How many solutions are needed to draw the graph?
A minimum of two solutions (two points) are needed to draw a straight line. A third point can be found for verification.
Q6. What is the difference between solutions in real numbers and natural numbers?
In real numbers, there are infinitely many solutions. In natural numbers (positive integers), the solutions are finite and limited because both x and y must be positive whole numbers.
Q7. Can x = 0 or y = 0 be part of a solution?
Yes. Setting x = 0 gives the y-intercept, and setting y = 0 gives the x-intercept. Both are valid solutions.
Q8. What does each solution represent geometrically?
Each solution (x, y) represents a point on the Cartesian plane. All solutions together form a straight line.
Q9. Can a linear equation in two variables have exactly one solution?
No. A single linear equation in two variables always has infinitely many solutions. A unique solution is obtained only when there are two independent linear equations (a system of equations).
Q10. Is this topic in the CBSE Class 9 syllabus?
Yes. Solutions of linear equations in two variables are part of the CBSE Class 9 Mathematics syllabus under the chapter Linear Equations in Two Variables.
